Classifying Real Numbers

Classifying Real Numbers

Numbers can be classified into sets based on their characteristics. Here are the five number sets we will earn about in this unit:

• Real Numbers

• Rational Numbers

• Irrational Numbers

• Integers

• Whole Numbers

Numbers often fall into more than one set

Here are some examples of how to identify numbers by the sets they belong to

Fill in the Blank

Let's try to identify some numbers by the sets they belong to. Fill in the blank to make this statement true: A pitcher pitched 5 and 1/3 innings. The number of innings pitched is a real number and it is also a _______ number.

$$5\frac{1}{3}$$  

Explanation:

If a pitcher pitched 5 and 1/3 innings, the number of innings pitched would be a real number and a rational number because it can be written as a fraction. Any number that can be written as a fraction is rational.

Multiple Select

Let's consider the square root of 10. Check all the number sets that would apply to this number

$$\sqrt{10}$$  

• real number

• rational number

• irrational number

• whole number

Explanation:

By understanding number sets, you can identify whether statements about the relationships between sets are true or false.

• All irrational numbers are real numbers. True. Real numbers include all rational and irrational numbers, so all irrational numbers are real numbers.

• No rational numbers are whole numbers. False. A whole number can be written as a fraction with a denominator of 1, so every whole number is included in the set of rational numbers.

  
  

Multiple Choice

Tell whether the following statement is true or false:

All rational numbers are integers

• True

• False

Explanation

The statement "All rational numbers are integers" is false because rational numbers include decimals, fractions, and percents. Even though every integer is a rational number, not every rational number is an integer.

Multiple Choice

Tell whether the following statement is true or false:

Some irrational numbers are integers

• True

• False

Explanation

The statement "Some irrational numbers are integers" is false because irrational numbers are neverending, nonrepeating decimals. Every integer terminates as a decimal, so an integer cannot also be an irrational number.

Real Numbers can be used to identify real-world quantities. Here are some examples:

• The number of people in the class with blonde hair. This number would best be described as a whole number because the number of people in a class with blonde hair would be counted using whole numbers equal to or greater than zero.

• The amount of water left in a cup measured in milliliters. This amount would best be described as a rational number because the amount of milliliters in a cup could involve a decimal point that terminates. All terminating or repeating decimals are rational numbers.

Multiple Choice

Which number set would best describe the outside temperature in Alaska during the winter, measured in degrees Fahrenheit?

• Irrational number

• whole number

• integer

Explanation

The outside temperature in Alaska measured in degrees Fahrenheit would best be described as an integer because the temperature would be a whole number or it's opposite (negative).

Practice

Let's practice what we've learned about number sets. Feel free to use your notes to help you answer these questions.

Multiple Select

Check all number sets that apply to the number shown below.

$$\frac{7}{8}$$  

• Real number

• Rational number

• Integer

• Irrational number

Multiple Select

Check all number sets that apply to the number shown below.

$$\sqrt{19}$$  

• Real number

• Rational number

• Integer

• Irrational number

Multiple Select

Check all number sets that apply to the number shown below.

$$0.\overline{34}$$  

• Whole number

• Rational number

• Integer

• Irrational number

Multiple Select

Check all number sets that apply to the number shown below.

$$0.75$$  

• Whole number

• Rational number

• Integer

• Irrational number

Multiple Select

Check all number sets that apply to the number shown below.

$$\sqrt{36}$$  

• Whole number

• Rational number

• Integer

• Irrational number

Multiple Select

Check all number sets that apply to the number shown below.

$$-86$$  

• Whole number

• Rational number

• Integer

• Irrational number

Multiple Choice

Is the following statement true or false?

"All whole numbers are rational numbers"

• True

• False

Multiple Choice

Is the following statement true or false?

"No irrational numbers are whole numbers"

• True

• False

Multiple Choice

Which number set would most accurately describe the number of pets each students owns?

• Integer

• Irrational Number

• Whole Number

• Rational Number

Multiple Choice

Which number set would most accurately describe a person's shoe size?

• Integer

• Irrational Number

• Whole Number

• Rational Number

Open Ended

How would you describe an irrational number in your own words? Type your response below.